Quantitative Data Analysis: Univariate (cont

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Quantitative Data Analysis Univariate (contd)
Bivariate Statistics

• Neuman and Robson Chapter 11.

Research Data library at SFU http//www.sfu.ca/rd
l/
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Class Session Activities

• Quiz 2
• More on Univariate Statistics
• Begin Bivariate Statistics
• If time
• Hans Rosling on Using Empirical Research to
  Understand World Change
• http//www.youtube.com/watch?vhVimVzgtD6w
• Hans Rosling Let my data set change your mind
  set
• http//www.youtube.com/watch?vKVhWqwnZ1eMfeature
  related

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Recall Univariate Statistics

• Frequency distributions explore each variable in
  a data set, separately to see the pattern of
  responses
• Measures of central tendency of the values (mean,
  median, mode)
• Measure of variation or variation (range,
  percentile, standard deviation, z-scores) 

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Studying Frequency Distributions

• Raw Data     Obtain a printout of the raw data
  for all the variables.
• resembles a matrix, with the variable names
  heading the columns, and the information for each
  case or record displayed across the rows.
• Source (for next examples) http//www.csulb.edu/
  msaintg/ppa696/696uni.htm    

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Example Raw data for a study of injuries among
county workers (first 10 cases)

• Raw data is difficult to grasp, especially with
  large number of cases or records.

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To present the information in a more organized
format, start with univariate descriptive
statistics for each variable. Example The
variable Severity of Injury
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Frequency Distribution for Severity of Injury

• Obtain a frequency distribution of the data for
  the variable.
• Identify the lowest and highest values of the
  variable,
• Put all the values of the variable in order from
  lowest to highest.
• count the number of appearance of each value of
  the variable. This is a count of the frequency
  with which each value occurs in the data set.  

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Grouped Data

• Decide on whether the data should be grouped into
  classes.    
• Example The severity of injury ratings can be
  collapsed into just a few categories or groups.
• Grouped data usually has from 3 to 7 groups.
• There should be no groups with a frequency of
  zero (in this example, there are no injuries with
  a severity rating of 7 or 8).
•    Ways to construct groups
• equal class intervals (e.g., 1-3, 4-6, 7-9).
• Approximately equal numbers of observations in
  each group.
• Remember that class intervals must be both
  mutually exclusive and exhaustive.

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Caution Grouping Response Categories

• To make new categories
• Facilitate analysis of trends
• But decisions have effects on the interpretation
  of patterns

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Cumulative Frequency Distributions

• include a third column in the table (this can be
  done with either simple frequency distributions
  or with grouped data
• How many injuries were at level 5 or lower?
  Answer7   

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Percentaged Frequency Distributions

•     Frequencies can also be presented in the form
  of percentage distributions and cumulative
  percentagescumulative percentages

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Why Graph?

• way of visually presenting data
• present the data
• summarize the data
• enhance textual descriptions
• describe and explore the data
• make comparisons easy
• avoid distortion
• provoke thought about the data

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Bar Graphs (Bar Charts)

• to display frequency distributions for variables
  measured at the nominal ordinal levels.
• use the same width for all the bars with space
  between bars.
• label the parts of the graph, including the
  title, the left (Y) or vertical axis, the right
  (X) or horizontal axis, and the bar labels.

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Another Bar Graph
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Histograms

•  for interval and ratio level variables
• width of the bar is important, since it is the
  total area under the bar that represents the
  proportion of the phenomenon accounted for by
  each category
• bars convey the relationship of one group or
  class of the variable to the other(s).      

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Histogram example

• In the case of the counties employee injuries,
  we might have information on the rate of injury
  according to the number of workers in each county
  in State X.

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Grouping Categories (Histograms)

• If we group injury rates into three groups
• low rate of injury would be 0.0-1.9 injuries per
  1,000 workers
• moderate would be 2.0-3.9
• high would be 4.0 and above (in this case, up to
  5.9).

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Frequency Polygon

• another way of displaying information for an
  interval or ratio level variable.
• also used to show time series graphs, or the
  changes in rates over time.

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Graph of Frequency Distribution (Univariate)
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Pie Chart

• Another way to show the relationships between
  classes or categories
• each "slice" represents the proportion of the
  total phenomenon that is due to each of the
  classes or groups.

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Another visual representation of a distributions
Pie charts
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Bivariate Statistics (relations between 2
variables)

• After examining univariate frequency distribution
  of the values of each variable separately,
• To study joint occurrence distribution of the
  values of the independent and dependent variable
  together.
• The joint distribution of two variables is called
  a bivariate distribution.    

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Contingency Tables (Cross-tabulations)

•    
• A contingency table shows the frequency
  distribution of the values of the dependent
  variable, given the occurrence of the values of
  the independent variable.
• Both variables must be grouped into a finite
  number of categories (usually no more than 2 or 3
  categories) such as low, medium, or high
  positive, neutral, or negative male or female
  etc.

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Features of Contingency Table

• Title
• Categories of the Independent Variable head the
  tops of the columns
• Categories of the Dependent Variable label the
  rows
• Order categories of the two variables from lowest
  to highest (from left to right across the
  columns from top to bottom along the rows).
  (Usually but not always).
• Show totals at the foot of the columns

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Basic Terminology (Tables)

• Parts of a Table
• title (conventions)
• Order of naming of variables
• Dependent, independent, control
• body, cell, column, row
• marginals
• sources, date

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Bivariate Statistics Parts of the Table
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Constructing a Contingency Table

•  if the variables not divided into categories,
  decide on how to group the data.    
• obtain a frequency distribution for the values of
  the independent variable
• obtain a frequency distribution for the values of
  the dependent variable
• obtain the frequency distribution of the values
  of the dependent variable, given the values of
  the independent variable (either by tabulating
  the raw data, or from a computer program
• display the results of step 4 in a table

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Table 1. Attitudes toward Consolidation by Area
of Residence

• Interpreting a Contingency Table
• Inspect the contingency table for patterns.
  (difficult if there are different totals of
  observations in the different categories of the
  independent variable)

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Interpreting a Contingency Table

• Convert the observations in each cell to a
  percentage of the column total
• be sure to still show the total number of
  observations for each column on which the
  percentages are based. (N total number per
  column)
• Compare the percentages across the categories of
  the dependent variable (the rows).

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Percentaged Contingency Table (example)Table 1b
Attitudes toward Consolidation by Area of
Residence
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Interpreting a Contingency TableTable 1.
Attitudes toward Consolidation by Area of
Residence

• more city residents (54) than non-city residents
  (37) are for consolidation. Conversely, more
  non-city residents (39) than city residents
  (19) are against consolidation. About the same
  percentage of both groups have no opinion about

Description More city residents (54) than
non-city residents (37) are for consolidation.
Conversely, more non-city residents (39) than
city residents (19) are against consolidation.
About the same percentage of both groups have no
opinion about consolidation.
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Grouping categories (Collapsing categories) U.N.
example
Babbie, E. (1995). The practice of social
research Belmont, CA Wadsworth
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Collapsing Categories omitting missing data
Babbie, E. (1995). The practice of social
research Belmont, CA Wadsworth
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Types of Relationships or Associations between
two variables

• Correlation (or covariation)
• when two variables vary together
• a type of association
• Not necessarily causal
• Can be same direction (positive correlation or
  direct relationship)
• Can be in different directions (negative
  correlation or indirect relationship)
• Independence
• No correlation, no relationship
• Cases with values in one variable do not have any
  particular value on the other variable

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What is an association between two variables?

• Can the value of one variable be predicted, if we
  know the value of the other variable?    
• Example half the people participating in
  training programs get a job. What is the
  likelihood of any one participant getting a job?
  About fifty-fifty. So we would not be very good
  at predicting whether people will get jobs or
  not.
• If we introduce a second variable (i.e. length of
  time in training), does it help us to be more
  accurate in our predictions of the likelihood
  that someone will get a job?

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Two variables

• Dependent variable Obtaining a Job No job100
  Gets a job100
• Independent Variable Length of Training Program
  Short100 Long100

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Bivariate Distribution--Perfect Positive
Relationship(If training is good for getting a
job)

• If we know the length of the training program,
  we can perfectly predict the likelihood of
  getting a job. The longer the training program,
  the more likely the participant is to get a job
  and, conversely, the shorter the training program
  the less likely the participant is to get a job.

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Bivariate Distribution--Perfect Inverse
Relationship

• If we know the length of the training program, we
  can perfectly predict the likelihood of getting a
  job. The longer the training program, the less
  likely the participant is to get a job and,
  conversely, the shorter the training program the
  more likely the participant is to get a job. That
  is, as the training program length increases,
  likelihood of obtaining a job decreases.

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Bivariate Distribution--No Relationship

• (If training has no relationship with getting a
  job)

50/50 guess. Knowing the length of the training
program does not help to predict the likelihood
of getting a job.
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Techniques for examining relationships between
two variables

• Cross-tabulations or percentaged tables
• Graphs, scattergrams or plots
• Measures of association (e.g. correlation
  coeficient, etc.)

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Scattergram (Bivariate)
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Interpreting a Relationship between two variables

• Do the patterns in the tables mean that there is
  a relationship between the two variables (in
  example area of residence and attitude toward
  consolidation)?
• Is one's attitude about consolidation associated
  with one's area of residence?
• If there is a relationship, how strong is it? Are
  the results statistically significant? Are the
  results meaningfully significant?
• In order to answer these questions, we must turn
  to a set of statistics called Measures of
  Association (next day).